If you have a quantum superposition of multiple states, can you tell without collapsing it what the component states are? I'm looking at stuff about quantum computing, and I only have really surface level knowledge but I find it interesting and I was trying to learn the programming language to write programs that can be run on a quantum computer.
I was thinking about making a program that creates a superposition of all the solutions to a problem.  But I do not know how to get all of the solutions out of this superposition.  It seems like you can obtain just one component, or basis state, out of the superposition if you measure the superposition and collapse it, destroying the superposition so you can't get any more solutions from it. However, I would like to get all of the solutions out, not just one at random.
If there is no way to get multiple results out, then I was also thinking of the user inputting one guess and the program tells you whether it is a solution or not. Is it possible, with or without collapsing the superposition, to check whether something is a solution to the problem?  This would be better than getting a solution at random.
In summary:
If you generate a mixed quantum state, can you determine what all of the basis states are, with or without collapsing the superposition? If not, then can you determine whether one particular state is one of the basis states, with or without collapsing the superposition?
 A: Just to clarify one point:

If you generate a mixed quantum state, can you determine what all of the basis states are, with or without collapsing the superposition?

A mixed state is a statistical ensemble of pure states. And a superposition refers to a state that can expressed as sum of other possible states - a pure state. You are referring to pure states I believe.
To answer this question, if you really do not know what the possible outcomes of measurement are, then no.
After you prepare a qbit, to determine all of its possible outcomes, you would need to measure it in many directions many times and under all conditions you are satisfied with. From this, you would obtain a pattern and therefore a probability distribution. This is what allows you to then express the original prepared bit as a superposition of other states.

If not, then can you determine whether one particular state is one of the basis states, with or without collapsing the superposition?

If you truly have no statistical information about the likelihood of possible outcomes, or what these outcomes may even be, without measuring the system (collapsing the state) as described above, then you cannot.
